Optimal. Leaf size=77 \[ \frac {10 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{21 b}-\frac {10 \cos (2 a+2 b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{7 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4385, 2716,
2720} \begin {gather*} \frac {10 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{21 b}-\frac {10 \cos (2 a+2 b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{7 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2720
Rule 4385
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx &=-\frac {\csc ^2(a+b x)}{7 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {10}{7} \int \frac {1}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {10 \cos (2 a+2 b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{7 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {10}{21} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {10 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{21 b}-\frac {10 \cos (2 a+2 b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{7 b \sin ^{\frac {3}{2}}(2 a+2 b x)}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 66, normalized size = 0.86 \begin {gather*} \frac {40 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\left (-13 \csc ^2(a+b x)-3 \csc ^4(a+b x)+7 \sec ^2(a+b x)\right ) \sqrt {\sin (2 (a+b x))}}{84 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 224.97, size = 154, normalized size = 2.00
method | result | size |
default | \(\frac {\sqrt {2}\, \left (-\frac {16 \sqrt {2}}{7 \sin \left (2 x b +2 a \right )^{\frac {7}{2}}}+\frac {8 \sqrt {2}\, \left (5 \sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \EllipticF \left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\sin ^{3}\left (2 x b +2 a \right )\right )+10 \left (\sin ^{4}\left (2 x b +2 a \right )\right )-4 \left (\sin ^{2}\left (2 x b +2 a \right )\right )-6\right )}{21 \sin \left (2 x b +2 a \right )^{\frac {7}{2}} \cos \left (2 x b +2 a \right )}\right )}{16 b}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.55, size = 177, normalized size = 2.30 \begin {gather*} -\frac {20 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + 20 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - \sqrt {2} {\left (20 \, \cos \left (b x + a\right )^{4} - 30 \, \cos \left (b x + a\right )^{2} + 7\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{84 \, {\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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